Optimizing a computational method for length lower bounds for reflecting sequences

We refine and optimize a computationally intensive enumeration method, based on the traversal of a quadtree, for finding lower bounds on the lengths of reflecting sequences for labeled chains. The improvement results from the introduction of a redundancy relation defined on vertex-pairs of the underlying quadtree, which enables the pruning of redundant branches near the root of the quadtree, as well as locally at deeper depths. The test run of the implementation showed a length lower bound of 19t - 214 for t-reflecting sequences for labeled 7-chains with significant speedup, which yields the current length lower bound Omega(n(1.51)) for universal traversal sequences for 2-regular graphs of n vertices, and Omega(d(2-1.51)n(2.51)) for universal traversal sequences for d-regular graphs of n vertices, where 3 less than or equal to d less than or equal to n/17 + 1.